Arithmetic Sensing III — Optimized Quadrature and Unavoidable Aliases

For midpoint times t_j=(j+1/2)T/m, the order-H cosine family uses w_j(c)=m^-1[1+2 sum(c_r cos(2 pi r t_j/T))]. Midpoint cosine sums preserve total mass one. Requiring the bracketed density to be nonnegative makes the weights a genuine positive quadrature rather than a signed cancellation device.

The centered response is real and affine in the coefficients. The finite design therefore exposes conditioning, density and arithmetic leakage as explicit constraints rather than as informal window folklore.

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Let arbitrary complex weights satisfy sum_j w_j=1. At every integer grid alias omega_q=2 pi q m/T, every midpoint has the same phase e^(i t_j omega_q)=(-1)^q. The common phase factors out of the weighted sum, so K_w(omega_q)=(-1)^q and |K_w(omega_q)|=1.

This result does not require positivity or symmetry. Weight optimization can suppress leakage before an alias, but it cannot impose a strict alias-height constraint on a fixed uniform grid. Alias location must be controlled jointly through m/T, and the remote tail must still be bounded analytically.

• Exact aliases are a structural property of the sampling schedule, not an optimization failure.
• A finite optimizer that ignores alias location can certify the wrong problem.

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For target frequencies Delta_nl=log(n/l), auxiliary variables bound the absolute centered responses. A Gershgorin constraint sum_{l != n} z_nl <= 1-gamma gives lambda_min(G_c) >= gamma. Sampled positivity and an upper density cap C keep the measurement positive without concentrating all weight on a few readings.

The divisor-weighted design tail N<k<=M_d is minimized by a linear program. When feasible, its optimum is the global minimum of that declared finite leakage proxy over the declared order-H cosine family, sampled positivity constraints, density cap and Gershgorin floor. It is not a claim of global optimality over all windows or schedules.

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After solving the finite program, the implementation discards its auxiliary variables and recomputes the exact realized kernel, exact target Gram spectrum, divisor-envelope correlations through a separate truncation, an analytic remainder beyond that truncation, the final coefficient bound and the weighted Gaussian sensor-noise covariance.

For the order-eight cosine window, the safe pre-alias endpoint is L_H=pi m/T-H(2 pi/T). The million-term certificate lies inside that endpoint for the reference parameters; the analytic remainder then controls the infinite tail, including the exact remote aliases that weighting cannot remove.

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At N=50, sigma=2, T=1,000 and m=5,000, the eight-harmonic tail-optimized point has minimum Gram eigenvalue 0.98182107, complete quadratic-tail bound 0.000232014 and effective sample count 2759.61. It is 117.17 times smaller than the re-audited Hann certificate at the same resources.

The balanced cap-2.0 point has complete bound 0.000495807 and is 54.83 times smaller than Hann. Reporting both points makes the conditioning/leakage/noise Pareto tradeoff visible.

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At 5,000 readings the optimized design certifies noiseless tail recovery, but the stated sensor noise sigma=0.01 is too large for the Gaussian union bound at that density. At 100,000 readings the computed failure bound is 6.09*10^-10. Tail leakage and sensor noise are separate resources.

A held-out quadratic-field experiment through norm 2,000 had maximum complex coefficient error 0.000002105 and integer-rounding success. This is an empirical falsification control, not part of the universal proof.

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The optimized weights do not reveal a new axiom of numbers. They are a measurement geometry adapted to a known arithmetic spectrum: logarithmic ratios of integers with a divisor-weighted risk. A small number of positive trigonometric corrections can exploit that structure more effectively than a generic window while displaying the conditioning and noise cost.

Arithmetic Sensing IV supersedes and sharpens the remote-alias part of this numerical certificate. It does not erase the exact-alias theorem; it supplies a stronger continuum-positive and alias-band audit.

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Proved here: exact unit-magnitude aliases for every weighting of the midpoint grid; affine centered responses; the finite-family linear program; Gershgorin conditioning from its constraints; and the declared alias-safe complete-tail certificate.

Computed and independently checked: global finite LP solutions, positive realized weights, exact Gram spectra, million-term certificates, density-cap Pareto points and held-out quadratic recovery.

Open: global optimality among all positive schedules, a closed-form extremal window, optimal joint scaling in N, T and m, and sharper all-alias bounds beyond the declared family.

• Floating-point optimization and spectral calculations are computations/certifications, not symbolic formal proofs.
• The active continuation is technical work, not a claim that the central Hann result was invalid.

Sources: [1] [4]